Long-Term Evolution (LTE) (see 3GPP. Evolved universal terrestrial radio access (e-utra); physical layer general description. Technical Specification 3GPP 36.201 Release-8, 3rd Generation Partnership Project, Sophia Antipolis, December 2009) is the trademark of the Third Generation Partnership Project (3GPP) and is aimed to become the next generation mobile network technology. In LTE, different transmission modes (TM) exist, see, for example, 3GPP. Evolved universal terrestrial radio access (e-utra); physical layer procedures. Technical Specification 3GPP 36.213 Release-8, 3rd Generation Partnership Project, Sophia Antipolis, December 2009, 3GPP. Evolved universal terrestrial radio access (e-utra); physical layer procedures. Technical Specification 3GPP 36.213 Release-9, 3rd Generation Partnership Project, Sophia Antipolis, March 2010, and 3GPP. Evolved universal terrestrial radio access (e-utra); physical layer procedures. Technical Specification 3GPP 36.213 Release-10, 3rd Generation Partnership Project, Sophia Antipolis, December 2010, which can be divided into open-loop and closed-loop transmission. In order to provide reliable communication, the user equipment (UE) has to estimate its channel. As indicated by the name ‘closed-loop’, the UE provides the eNodeB with information which is referred to as Channel State. Information (CSI). The CSI concerns the instantaneous state of the channel, see, for example 3GPP. Evolved universal terrestrial radio access (e-utra); physical channels and modulation. Technical Specification 3GPP 36.211 Release-8, 3rd Generation Partnership Project, Sophia Antipolis, December 2009, 3GPP. Evolved universal terrestrial radio access (e-utra); physical channels and modulation. Technical Specification 3GPP 36.211 Release-9, 3rd Generation Partnership Project, Sophia Antipolis, March 2010, and 3GPP. Evolved universal terrestrial radio access (e-utra); physical channels and modulation. Technical Specification 3GPP 36.211 Release-10, 3rd Generation Partnership Project, Sophia Antipolis, December 2010.
One parameter which may be fed back is the Precoding Matrix Index (PMI). Precoding vectors are a special kind of precoding matrices, wherein the precoding vectors are precoding matrices that only comprise a single column. The precoding matrices/precoding vectors are stored in a codebook and are known to both, eNodeB and UE, so it is sufficient to feed back only an index in order to save transmission bandwidth. In the following, reference is made to precoding vectors, while the concepts, explanations and teachings also apply to precoding matrices.
As already indicated, the precoding vector may be selected by the UE. Depending on the number of transmit antennas, several precoding vectors exist. It is a priori not known which of these precoding vectors maximizes the signal energy. According to the state of the art, at first, the signal energy is determined for all precoding vectors individually, and then, the signal energies of the precoding vectors are compared in order to feed back the most appropriate precoding vector index (or precoding matrix index, respectively). Such determination processes and comparisons are energy consuming and time consuming. It can be concluded that precious time and energy is wasted, which both are limited resources at each UE.
In the following, a complex baseband notation will be used, which deploys a matrix vector calculus for describing the system structure and the signal processing. Vectors and matrices may be denoted by lower case or upper case characters in bold face.
The matrix lk is the identity matrix of dimension k. Furthermore, (•)* and (•)H denote the conjugate and Hermitian operation, respectively. The magnitude of a scalar value is denoted by |•|, where the Euclidean norm ∥•∥2 will be used for vectors. The subscript of the Euclidean norm will be skipped in the following.
In the following, closed-loop transmission in LTE-Release 8, 9 or 10 systems using NT transmit antennas at eNodeB and NR receive antennas at UE is considered. The system model is defined by the formula:r=Hd+n  (1)and describes the transmission of symbol dε using the channel HεNR×NT using the precoding vector εNT. The UE receives rεCNR, where nεNR, n˜CN (0, σ2INR) is the zero-mean circularly-symmetric complex Gaussian noise vector.
According to the state of the art, the UE reports the index i of the precoding vector from the set of available precoding vectors, which yields maximal signal energy ∥Hpi∥2. In order to obtain the best PMI for M subcarriers, according to the state of the art, the following has to be calculated:
                              arg          ⁢                                          ⁢                                    max                              i                ∈                                  {                                      1                    ,                    …                    ⁢                                                                                  ,                    K                                    }                                                      ⁢                          {                                                ∑                                      j                    =                    0                                                        M                    -                    1                                                  ⁢                                                                                                                        H                        j                                            ⁢                                              p                        i                                                                                                  2                                            }                                      =                              arg            ⁢                                                  ⁢                                          max                                  i                  ∈                                      {                                          1                      ,                      …                      ⁢                                                                                          ,                      K                                        }                                                              ⁢                              {                                                      ∑                                          j                      =                      0                                                              M                      -                      1                                                        ⁢                                                            p                      i                      H                                        ⁢                                          H                      j                      H                                        ⁢                                          H                      j                                        ⁢                                          p                      i                                                                      }                                              =                                    arg              ⁢                                                          ⁢                                                max                                      i                    ∈                                          {                                              1                        ,                        …                        ⁢                                                                                                  ,                        K                                            }                                                                      ⁢                                  {                                                                                    p                        i                        H                                            ⁢                                              H                        0                        H                                            ⁢                                              H                        0                                            ⁢                                              p                        i                                                              +                    …                    +                                                                  p                        i                        H                                            ⁢                                              H                                                  M                          -                          1                                                H                                            ⁢                                              H                                                  M                          -                          1                                                                    ⁢                                              p                        i                                                                              }                                                      =                          arg              ⁢                                                          ⁢                                                max                                      i                    ∈                                          {                                              1                        ,                        …                        ⁢                                                                                                  ,                        K                                            }                                                                      ⁢                                  {                                                                                    p                        i                        H                                            ⁡                                              (                                                                                                            H                              0                              H                                                        ⁢                                                          H                              0                                                                                +                          …                          +                                                                                    H                                                              M                                -                                1                                                            H                                                        ⁢                                                          H                                                              M                                -                                1                                                                                                                                    )                                                              ⁢                                          p                      i                                                        }                                                                                        (        2        )                                                          ⁢                              =                          arg              ⁢                                                          ⁢                                                max                                      i                    ∈                                          {                                              1                        ,                        …                        ⁢                                                                                                  ,                        K                                            }                                                                      ⁢                                  {                                                                                    p                        i                        H                                            (                                                                                                    ∑                                                          j                              =                              0                                                                                      M                              -                              1                                                                                ⁢                                                                                    H                              j                              H                                                        ⁢                                                          H                              j                                                                                                                                ︸                                                      :=                            R                                                                                              )                                        ⁢                                          p                      i                                                        }                                                              ,                                    (        3        )            where K=2NT is the number of precoding vectors and wherein NT is the number of transmit antennas at eNodeB. According to the state of the art, an exhaustive search is to be performed by analyzing all possible precoding vectors and searching for the appropriate index. For the case of NT=2 transmit antennas and NT=4 transmit antennas, four and sixteen different precoding vectors exist, respectively.